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Properties

Label 1-69-69.26-r1-0-0
Degree $1$
Conductor $69$
Sign $0.948 - 0.316i$
Analytic cond. $7.41507$
Root an. cond. $7.41507$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (0.415 + 0.909i)10-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.654 + 0.755i)20-s + 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (0.415 + 0.909i)10-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.654 + 0.755i)20-s + 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(7.41507\)
Root analytic conductor: \(7.41507\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (1:\ ),\ 0.948 - 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.105959767 - 0.5050243420i\)
\(L(\frac12)\) \(\approx\) \(3.105959767 - 0.5050243420i\)
\(L(1)\) \(\approx\) \(2.046623429 - 0.2524746425i\)
\(L(1)\) \(\approx\) \(2.046623429 - 0.2524746425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.68794378856051732439975773976, −30.8313146629914164800710536232, −29.640768018923275613682431860869, −28.48208878463922248841316428509, −27.45429696087071405989660104139, −25.716342941688044725361256169992, −24.64407225263486454406990565614, −24.28968418460898364611238926480, −22.70266447613098790421923232048, −21.77293188191577397821646465822, −20.71451905289889515562456855323, −19.774657775798159487664487858751, −17.901502509252980528212966425815, −16.700380010145091990252628150052, −15.63760865248521206170149138926, −14.543516750462126774393332117841, −13.202488589093452485279512433399, −12.27481421969455419964871000242, −11.185118793315137380042784686053, −9.08405675765898490479352445757, −7.954111832177381724922097056203, −6.08758727867712356910578421930, −5.17288282486629889817257198988, −3.69524340556137472688961134057, −1.76837290210272998815939664341, 1.69434729187003615187431376139, 3.414720688779706155299095060221, 4.61903730925202056701812359956, 6.46890371648675892092528265441, 7.24316781799658016943887991493, 9.60585987186587920179202235624, 11.0008668711561666831100400656, 11.66283330113456578517806496852, 13.53057073513273044724802604653, 14.16081982326472893470047906867, 15.25095423120145064974561776389, 16.69693468509757276975778258654, 18.15698408967814820601094163590, 19.53825722947081625916674454159, 20.48166337425270576607460709492, 21.78037557290309072043864077058, 22.603067184802118337674689935416, 23.60670745144030788943031712398, 24.69761248140389609878816996892, 26.0170686873378005297225042983, 27.09815001923394904749051944879, 28.61046194954905964920256635092, 29.71282506376405823257237174442, 30.493884108297899109516800129707, 31.210449606202613939568404831272

Graph of the $Z$-function along the critical line